Ans should be an option (A)
For a quadratic equation to have real and distinct roots, it’s discriminant should be strictly greater than zero.
$\therefore$ $b^{2}-4ac\gt0$
$\Rightarrow$ $16-4(-\log_{2}A)\gt0$ $\Rightarrow$ $\log_{2}A\gt-4$ $\Rightarrow$ $A\gt \frac{1}{16}$